Frazioni multiple graduate e divisioni tra interi

Liber abaci(V.1.2) 
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

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denominans
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-.".
Liber
)*/
))) 

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b
2
b
3
...
a
n
b
1
b
2
b
3
...b
n
))
b

6b
73
6***6 b
3
*
"")))".
'(*28"
.')"(*
2
#""
"regolab)"
*
1"9*/
":;") 3,*/");
3:)"%"
)&3,
*4)";.
;*!
*
/"
3
1Enrico Giusti e altri, Un ponte sul mediterraneo. Leonardo pisano, la scienza araba e la rinascita della matematica in
Occidente, Polistampa, 2002
a
2 
a
1
b
2 
b
1
a
1
b
1
a
2
b
1
b
2
1
b
n
b
n1
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1
10...0
b
n
b
n1
...b
1
Nei trattati d’abaco successivi, dal trattato di Aritmetica di Paolo dall’Aaco, ca 1350, no al Trattato
d’Aritmetica di Giseppe Maria Figatelli, pubblicato nel 1664 e ristampato più volte per tutto il Settecento, il
procedimento di passaggio dalla frazione multipla a quella ordinaria si trova sotto il nome di “inlzare i
rotti”. Il passaggio inverso, dalla frazione ordinaria alla multipla (o come si diceva ai rotti in lza, cioè ‘in la’)
è chiamato dal Figatelli “traslatare i rotti”
Traslatando i rotti)
<" 
'inlzare i rotti("
)*-"ab
,

 b
3

".
a=b
,
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,
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,
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,
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,
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,
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3
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3
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3
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3
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,
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3
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3
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,
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3
b
,
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,
a
3
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,

"
?6a
3
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?6a
,
6b
,
!q$""
ab
,
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a
1
b
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b
1

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1
63

b
,
a
3
>a
,
6b
,
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3
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,
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,
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3
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,
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3
1 ),)
*/)
*
5")@

".#
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1
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))qA.#>'() 
a
3
a
,
b
3
b
,
*
0"
*+
B",?3,*/a
 )
"*+
C:"D
C:D*1))
",&)E?)E?
)#
,)C")C D
)D *Traslatando"
&,D*
/#".")
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""
)),*
!))
)traslatando i rotti,
a
3

)))a
,

#
"*
a
2 
a
1
b
2 
b
1
q
a
240
95
1220
7
97
6024
2 2
7
24
9
24 60
a
n
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3
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2
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1
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3
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2
b
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1
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1
a
2
b
1
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2
a
3
b
1
b
2
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3
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a
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b
1
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2
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n
F#)a

6b

)
*/a

6b
a
73
6b
73
 *** a
;73
6b
;73
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;
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;
*4"
a
;
b
;
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a
;
= b
;
q >r
;
?6r
;
6b
;
*
-
!;7
))*1
#)"""*
0)"*-#
.3,)3,
3,)
.,?),?
.,?*
-)"
a

6 b

)
0"=3*-"
""")*
0#))
"a
>3
6 b
>3
a < b#"*
Ha
>3
>a b
>3
6b
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>a b
>3
=3>a b
>3
6bb
>3
*
+)'(
)"))
""* /
a
1
b
1
a
2
b
1
b
2
a
3
b
1
b
2
b
3
...
a
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b
1
b
2
b
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a
1
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a
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a
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a
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a
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k
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a
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1
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*)""
b
6b
73
6***6 b
,
6b
3
#))
#))"".*!
.#"""
.".*
!""b
b
73
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,
b
3
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""b
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".*/)"
""""
.I""".*!")"
)*!
"")*
F)#)(V.44)"a.
b
*)"b
73
 b
7,

*
a = a
b
+ r
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a
= a
73
b
73
+ r
73
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73
< b
73
a
73
= a
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b
7,
+ r
7
 ?< r
7,
< b
7,
a
,
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3
b
,
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,
 ?< r
,
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,
a
3
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b
3
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3
 ?< r
3
< b
3
F#"""""
a
?
 
"a *0#)J#

.)"
3
K
3

#"*
a
b
n
b
n1
...b
2
b
1
r
n
r
n1
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2
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1
b
n
b
n1
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2
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1
a
0
r
n
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n1
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1
b
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b
n1
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2
b
1
b
n
b
n1
...b
2
b
1
"))"ECLDL3CE?*-J
regola3CE?
3CE?=,
M
L
M
3?
M
33)"'(ECLDL
 *
67898
1760
=
0 5 3 6
2 8 10 11
38
+"
67898
1760
= 38 +
6
11
+
3
110
+
5
880
")))"

38 +
6
11
<
67898
1760
< 38 +
7
11
/")"""
traslatare i rottiE
M
3E?>%
M
3E>3?=3?3L"%L
3?3LK3CE?,#evitatione
"".)""#
 ":?DKLL?)"
*
9))
"N))
*2#I
)
")"
67898
2 8 10 11
#""
*
F)3L "C&DC:*
-
D)

/",%:)*
!)"
4"C&DC:
Divisione di 749 per 75
2)"N)
Liber abaci3L:C
)":""C&DC:
Poiché si sa come trovare le regole
2
dei numeri, qualcuno avrà voluto dividere 749 per
75,
,Nella traduzione abbiamo omesso 5 nel testo che, anche consultando altri studiosi, ci è sembrato un errore.
)C:
trovata la regola di 75, che è
1 0 0
3 5 5
"C&D%O,&D,,%")
"*
si divida 749 per 3, farà 249, e resta 2; questo 2 si ponga sopra il 3 della linea di
frazione conservata da parte
",&D:%")O&D&
&:
e si divida 249 per 5, per quello cioè che antecede il 3 in frazione, farà 49 e resta 4;
questo 4 si ponga sora lo stesso 5,
&D":")OD&&
:D
si divida di nuovo 49 per 5, per quello che si trova alla ne della linea di frazione, fa 9 e
rimane 4; questo 4 si ponga sopra lo stesso 5, e 9 si ponga
")O"
2 4 4
3 5 5
9
*
davanti alla medesima linea di frazione; e così di avrà dalla divisione richiesta , come qui
si mostra,
2 4 4
3 5 5
9
*
F'(*
-J)"
"C&DC:C:C:=%O:O:

1
3×5×5
=
1 0 0
3 5 5
4"C&D%C&D=,&DO%>,*/",
%*
2 0 0
3 5 5
",&D:)%

,&D=&DO:>&"&:
2 4 0
3 5 5
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5000
7000
2417
7101010
7
10
1
100
4
1000
2
7000